Chapter 2 Fundamentals of Seizure Analysis Algorithm
2.2 Chaotic Modeling for EEG
2.2.1 Introduction to Chaos Theorem
Recall that a Newtonian deterministic system is a system whose present state is fully determined by its initial conditions [4] (at least, in principle), in contrast to a stochastic (or random) system, for which the initial conditions determine the present state only partially, due to noise or other external circumstances beyond our control.
For a stochastic system, the present state reflects the past initial conditions plus the particular realization of the noise encountered along the way. So, in view of classical science, we have either deterministic or stochastic systems.
For a long time, scientists avoided the irregular side of nature, such as disorder in a turbulent sea, in the atmosphere, and in the fluctuation of wild–life populations.
Later, the study of this unusual results revealed that irregularity, nonlinearity, or chaos was the organizing principle of nature.
A modern scientific term deterministic chaos depicts an irregular and unpredictable time evolution of many (simple) deterministic dynamical systems, characterized by nonlinear coupling of its variables. Given an initial condition, the dynamic equation determines the dynamic process, i.e., every step in the evolution.
However, the initial condition, when magnified, reveals a cluster of values within a certain error bound. For a regular dynamic system, processes issuing from the cluster
are bundled together, and the bundle constitutes a predictable process with an error bound similar to that of the initial condition. In a chaotic dynamic system, processes issuing from the cluster diverge from each other exponentially, and after a while the error becomes so large that the dynamic equation losses its predictive power.
For example, in 1960s, Ed Lorenz from MIT created a simple weather model in which small changes in starting conditions led to a marked changes in outcome, called sensitive dependence on initial conditions, or popularly, the butterfly effect (i.e., “the notion that a butterfly stirring the air today in Peking can transform storm systems next month in New York, or, even worse, can cause a hurricane in Texas”). Thus long–range prediction of imprecisely measured systems becomes impossibility.
The character of chaotic dynamics can be illustrated with the logistic map as follows [5] :
1 (1 )
n n n
x + =rx −x , (2.1)
a discrete-time analog of the logistic equation for population growth. Here, xn ≥ is 0 a dimensionless measure of the population in the nth generation, and r≥0 is the intrinsic growth rate. We restrict the control parameter r to the range 0≤ ≤r 4 so that (2.1) maps the interval 0≤ ≤x 1 into itself.
A. Period-Doubling
Suppose we fix r, choose some initial population x0, and then use (2.1) to generate the subsequent xn. For the growth rate, we can consider cases as follows.
(1) When r< , the population always goes extinct: 1 xn → as 0 n→ ∞. (2) When 1< <r 3 the population grows and eventually reaches a non-zero
steady state, called a period-1 cycle.
(3) Table 2-1 shows the results of logistic map of initial condition x0 =0.4 and x0 =0.8, and we can find that even though there is a huge difference between the initial conditions, the two series as shown in Fig. 2-4 converge
to the same value in a moment.
Table 2-1 Results of logistic map (r=2.8) Logistic growth equation (r=2.8)
X(0) X(1) X(2) X(3) … X(22) X(23) X(24) X(25) 0.4 0.672 0.617 0.661 … 0.642 0.643 0.642 0.642 0.8 0.448 0.692 0.596 … 0.643 0.642 0.643 0.642
Fig. 2-4 Logistic map of period-1 cycle
(4) When r=3.14, the population builds up again but now oscillates about the former steady state, alternating between a large population in one generation and a smaller population in the next. This type of oscillation, in which xn repeats every two iterations, is called aperiod-2 cycle. Table 2-2 shows the results of logistic map of initial condition x0 =0.1 and x0 =0.3, and the series as shown in Fig. 2-5 reach the same two states after a while.
Table 2-2 Results of logistic map (r=3.14) Logistic growth equation (r=3.14)
X(0) X(1) X(2) X(3) … X(22) X(23) X(24) X(25) 0.1 0.2826 0.6365 0.7264 … 0.5385 0.7803 0.5382 0.7804 0.3 0.6594 0.7050 0.6527 … 0.7792 0.5402 0.7799 0.5389
Fig. 2-5 Logistic map of period-2 cycle
Further period-doublings to cycles of period 8, 16, 32, . . . , occur as r increases.
Specifically, let rn denote the value of r where a 2n-cycle first appears. Then computer experiments reveal that r1 = , 3 r2 =3.449, r3 =3.54409, …, r∞ =3.5699. The convergence is essentially geometric: in the limit of large n, the distance between successive transitions shrinks by a constant factor (2.2).
1
In fact, the same convergence rate appears no matter what unimodal map is iterated. In this sense, the number δ is universal. It is a new mathematical constant, as basic to period-doubling as n is to circles.
When r r> , the answer turns out to be complicated: For many values of r , ∞ the sequence
{ }
xn never settles down to a fined point or a periodic orbit instead thelong-term behavior is aperiodic. Table 2-3 shows the results of logistic map. It is interesting to note that even though the difference of the initial conditions is only 0.0001, the two series as shown in Fig. 2-6 are totally divergent in a minute.
Table 2-3 Results of logistic map (r=3.9) Logistic growth equation (r=3.9)
X(0) X(1) X(2) X(3) … X(22) X(23) X(24) X(25) 0.4 0.936 0.23362 0.6982 … 0.9184 0.2919 0.8062 0.609 0.4001 0.93607 0.23336 0.6977 … 0.6759 0.8542 0.4856 0.9741
Fig. 2-6 Logistic map of r=3.9
A bifurcation diagram summarizes the above phenomenon (Fig. 2-7). The horizontal axis shows the values of the parameter r while the vertical axis shows the possible values of x.
At r=3.4, the attractor is a period-2 cycle, as indicated by the two branches.
As r increases, both branches split simultaneously, yielding a period-4 cycle. This splitting is the period-doubling bifurcation mentioned earlier. A cascade of further period-doublings occurs as r increases, yielding period-8, period-16, and so on, until at r r= ∞ ≈3.57, the map becomes chaotic and the attractor changes from a
finite to an infinite set of points.
Fig. 2-7 Bifurcation diagram for the Logistic map
B. Basic Terms of Nonlinear Dynamics
Recall that nonlinear dynamics is a language to talk about dynamical systems.
Here, brief definitions are given for the basic terms of this language [4].
z Dynamical system: A part of the world which can be seen as a self–contained entity with some temporal behavior. Mathematically, a dynamical system is defined by its state and by its dynamics.
z Phase space: In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space.
z Attractor: An attractor is a ‘magnetic set’ in the system’s phase space to which all neighboring trajectories converge. More precisely, we define an attractor to be a subset of the phase space with the following properties:
(1) It is an invariant set;
(2) It attracts all trajectories that start sufficiently close to it;
(3) It is minimal (it cannot contain one or more smaller attractors).
A strange attractor shown in Fig. 2-8 is defined to be an attractor that exhibits sensitive dependence on initial conditions. Geometrically, an attractor can be a point, a curve, a manifold, or even a complicated set with a fractal structure known as a strange attractor.
Fig. 2-8 A plot of Lorenz's strange attractor
z Fractal: Roughly speaking, fractals are complex geometric shapes with fine structure at arbitrarily small scales. Usually they have some degree of self-similarity. In other words, if we magnify a tiny part of a fractal, we will see features reminiscent of the whole. Sometimes the similarity is exact;
more often it is only approximate or statistical.
Fractals are of great interest because of their exquisite combination of beauty, complexity, and endless structure. They are reminiscent of natural objects like mountains, clouds, coastlines, blood vessel networks, and even broccoli, in a
way that classical shapes like cones and squares can't match.
z Embedding dimension: The number of variables needed to characterize the state of the system. Equivalently, this number is the dimension of the phase space.
z Fractal dimension: The strange attractors typically have fractal microstructure. The attractor dimension counts the effective number of degrees of freedom in the dynamical system, described by a non integer dimension.
C. The link between EEG and chaos
Within the context of brain dynamics [4], there are suggestions that “the controlled chaos of the brain is more than an accidental by–product of the brain complexity” and that “it may be the chief property that makes the brain different from an artificial intelligence machine”. Namely, Chaos drives the human brain away from the stable equilibrium, thereby preventing the periodic behavior of neuronal population bursting.
The EEG, being the output of a multidimensional system [6], has statistical properties that depend on both time and space. Components of the brain (neurons) are densely interconnected and the EEG recorded from one site is inherently related to the activity at other sites. This makes the EEG a multivariable time series. The analysis of such nonlinear dynamical systems from time series involves state space reconstruction, and we will introduce in the next section.
If prediction becomes impossible, it is evident that a chaotic system can resemble a stochastic system, say a Brownian motion. However, the source of the irregularity is quite different. For chaos, the irregularity is part of the intrinsic dynamics of the system, not random external influences. Usually, though, chaotic
systems are predictable in the short–term. This short–term predictability is useful.
Chaos theory has developed special mathematical procedures to understand irregularity and unpredictability of low–dimensional nonlinear systems. Lyapunov exponents and attractor dimension are some examples. Lyapunov exponents evaluate the sensitive dependence to initial conditions estimating the exponential divergence of nearby orbits, and we will discuss the method later. Correlation dimensions estimate the fractal dimension and will be described in Chapter 3.